U.S. Secretary of State John Kerry said Sunday after four hours of talks with Russian Foreign Minister Sergey Lavrov that Russia had reaffirmed its commitment to finding a diplomatic solution in Ukraine but had not agreed to move Russian troops from the Ukraine border.

"We both made suggestions as to how that will be achieved ... and I will return to Washington to consult with President Obama on his choices," Kerry said at a news conference in Paris. "We are trying to find a way to defuse this."

Kerry said Lavrov indicated Russia "wants to support" Ukraine in its move toward independence but said the massing of Russian troops has created "a climate of fear and intimidation."

"Is it smart at this moment in time to have that number of troops amassed on a border when you are sending a message that you want to de-escalate and move in the other direction?" Kerry said.

Washington (CNN) - Some U.S. lawmakers are ready to say that it's futile to try to persuade Russia to give up control of Crimea.

Sen. Dianne Feinstein, chairwoman of the Senate Intelligence Committee, told CNN chief political correspondent Candy Crowley on Sunday that the debate over the Crimean Peninsula is "done" and the region is now under Moscow's control.

Tensions between the United States and Russia over the crisis in Crimea have exploded into an open row as Russia rejects U.S. diplomatic efforts to solve the impasse.

Russian Foreign Minister Sergey Lavrov said Monday that U.S. Secretary of State John Kerry postponed a face-to-face meeting with Russian President Vladimir Putin to discuss American proposals, which Moscow has effectively rejected, on solving the crisis.

The meeting, which Russia said was supposed to happen Monday, would have marked the highest-level contact between the two countries since Russian troops took up positions in Crimea, and would have come ahead of Sunday's potentially explosive vote on whether Crimea should split from Ukraine and join Russia.

The names involved in the escalating crisis in Ukraine - Crimea, Sevastopol, Simferopol - make it sound far away and far removed from America's allies and its interests. But the tensions there in fact hit very close to home for the United States.

First, Ukraine is not a distant empire but an integral part of Europe. Its capital, Kiev, is just a short flight from cities Americans visit all the time: Rome, Frankfurt, Paris.

Senior Senate Democrats, including Majority Leader Harry Reid, cautioned Monday that any U.S. move toward sanctions against Russia over Ukraine should be carried out in accordance with European allies.

Separately, Republican Ron Johnson, a member of the Senate Foreign Relations Committee, said he's urging fellow lawmakers to speak with "one voice" as they discuss action over the developing situation in Crimea.

Russian military exercises near Ukraine are raising concerns that Moscow may be putting troops in position to move across the border if such orders are issued, a senior U.S. official familiar with the most recent administration assessment told CNN Thursday.

But the United States still believes that Russia doesn't plan to order its forces into its tumultuous neighbor, the official said.

U.S. officials - who are monitoring the area 24 hours a day - have not yet seen signs that Russia is preparing to secure supply and transportation routes that would be crucial to any such movement, the official said.

Russian military activity levels observed by the United States also "appear to be within normal range," the official said.

President Barack Obama and Russian President Vladimir Putin went to great lengths to portray their phone call Friday as evidence the two leaders were working together to stabilize the situation in Ukraine.

Absent were the accusations they traded the past two months over the future of the former Soviet republic.

The Obama administration continues to dismiss the notion of a new Cold War with Russia. But the louder their protests, the more apparent the chill has become.

It was on display Saturday in widely differing characterizations of a telephone conversation between Secretary of State John Kerry and his Russian counterpart, Sergey Lavrov, after news broke that Ukrainian President Viktor Yanukovych left the capital because of what he described as a "coup."

Gay rights, Edward Snowden, Syria and now Ukraine: They're all recent issues in which the United States and Russia have had disagreements.

Tension has always seemed to exist between the two countries, and that's certainly been the case for President Obama and Russian President Vladimir Putin. Some days, it almost seems like the Cold War never went away.

In the latest example, Ukraine, it appears the Russian-backed government of President Viktor Yanukovych has been removed from office after deadly protests, setting up a power vacuum in a country known for Russian meddling.

The U.S. stands with Ukrainian opposition forces hoping to increase democratic reforms and decrease influence from Moscow, while Russia slams the opposition, saying they failed to honor international agreements made last week aimed at ending the crisis.

President Barack Obama and Russian President Vladimir Putin "exchanged views" on the need to quickly implement the political agreement reached on Friday in Ukraine and for all sides to "refrain from further violence."

A White House statement said the two leaders spoke by phone and also discussed Syria, including the "importance of efforts to advance a political solution" to the civil war.

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CNN's Security Clearance examines national and global security, terrorism and intelligence, as well as the economic, military, political and diplomatic effects of it around the globe, with contributions from CNN's national security team in Washington and CNN journalists around the world.